Chapter 3 Integers and rationals

3.1 The integers

Definition 3.1 An integer is an expression of the form $a\!\!-\!\!-b$, where $a$ and $b$ are natural numbers. Two integers are considered to be equal, $a\!\!-\!\!-b = c\!\!-\!\!-d$, iff $a+d = c+b$. We let $\mathbb{Z}$ denote the set of all integers.

Definition 3.2 The sum of two integers, $(a\!\!-\!\!-b) + (c\!\!-\!\!-d)$, is defined by the fomula \[ (a\!\!-\!\!-b) + (c\!\!-\!\!-d):=(a+c)\!\!-\!\!-(b+d) \] The product of two integers, $(a\!\!-\!\!-b) \times (c\!\!-\!\!-d)$, is defined by \[ (a\!\!-\!\!-b) \times (c\!\!-\!\!-d):=(ac+bd)\!\!-\!\!-(ad+bc) \]

One can verify that the addition and multiplication of integers are well-defined.

We can find that there is an isomorphism between the natural number $n$ and those integers of the form $n\!\!-\!\!-0$. That is, the integer $n\!\!-\!\!-0$ behave in the same way as the natural numbers $n$: \[ \begin{aligned} (n\!\!-\!\!-0)+(m\!\!-\!\!-0)&=(n+m)\!\!-\!\!-0;\\ (n\!\!-\!\!-0)\times(m\!\!-\!\!-0)&=nm\!\!-\!\!-0;\\ n\!\!-\!\!-0=m\!\!-\!\!-0& \iff n=m. \end{aligned} \] Thus we can identify the natural numbers with integers by setting $n \equiv n\!\!-\!\!-0$.

Definition 3.3 (Negation of integers) If $(a\!\!-\!\!-b)$ is an integer, we define the negation $-(a\!\!-\!\!-b)$ to be the integer $(b\!\!-\!\!-a)$.

One can check the definition is well-defined.

Lemma 3.1 (Trichotomy of integers) Let $x$ be an integer. Then exactly one of the following three statements is true: (a) $x$ is zero; (b) $x$ is equal to a positive natural number $n$; or (c) $x$ is the negation $-n$ of a positive natural number $n$.

Proposition 3.1 (Laws of algebra for integers) Let $x,y,z$ be integers. Then we have \[ \begin{aligned} x+y&=y+x\\ (x+y)+z&=x+(y+z)\\ x+0=0+x&=x\\ x+(-x) = (-x)+x&=0\\ xy&=yx\\ (xy)z&=x(yz)\\ x1=1x&=x\\ x(y+z)&=xy+xz\\ (y+z)x=yx+zx \end{aligned} \]

The above set of nine identities are asserting that integers from a commutative ring (If one deleted the identity $xy=yx$, then they would only assert that the integers form a ring).

Now we define the operation of substraction of two integers by the fomula \[ x-y:=x+(-y). \]

Proposition 3.2 (Integer have no zero divisors) Let $a$ and $b$ be integers such that $ab=0$. Then either $a=0$ or $b=0$ (or both).

Corollary 3.1 (Cancellation law for integers) If $a,b,c$ are integers such that $ac=bc$ and $c$ is non-zero, then $a=b$.

Definition 3.4 (Ordering of the integers) Let $n$ and $m$ be integers. We say that $n$ is greater than or equal to $m$, and write $n\geq m$ or $m\leq n$, iff we have $n=m+a$ for some natural number $a$.We say that $n$ is strictly greater that $m$, and write $n>m$ or $m<n$, iff $n\geq m$ and $n \ne m$.

Lemma 3.2 (Properties of order) Let $a,b,c$ be integers.

$a>b$ iff $a-b$ is positive natural number.
(Addition preserves order) If $a>b$, then $a+c>b+c$.
(Positive multiplication preserves order) If $a>b$ and $c$ is positive, then $ac>bc$.
(Negation reverses order) If $a>b$, then $-a<-b$.
(Order is transitive) If $a>b$ and $b>c$, then $a>c$.
(Order trichotomy) Exactly one of the statements $a>b$, $a<b$, or $a=b$ is true.

Here we consider the last one. First, if $a,b$ are both positive, then it has been proven in Chapter 1; If $a,b$ are both negative, since negation reverses order, we only need to consider $-a,-b$, which are both positive; as for other situation, we know that negative integers are small than zero, and zero is smaller than any positive integer.

Exercise 3.1 Show that the principle of induction does not apply directly to the integers.

3.2 The rationals

Definition 3.5 A rational number is an expression of the form $a//b$, where $a$ and $b$ are integers and $b$ is non-zero; $a//0$ is not considered to be a rational number. Two rational numbers are considered to be equal, $a//b=c//d$, iff $ad=cb$. The set of all rational numbers is denoted $\mathbb{Q}$.

Definition 3.6 If $a//b$ and $c//d$ are rational numbers, we defined their sum \[ (a//b) + (c//d) := (ad+bc)//(bd) \] their product \[ (a//b)\times(c//d) := (ac)//(bd) \] and the negation \[ -(a//b):=(-a)//b. \]

One can verify that the sum, product, and negation operations on rational numbers are well-defined.

We note that the rational numbers $a//1$ behave in manner identical to the integer $a$: \[ \begin{aligned} (a//1)+(b//1)&=(a+b)//1;\\ (a//1)\times(b//1)&=(ab)//1;\\ -(a//1)&=(-a)//1. \end{aligned} \] Because of this, we will identify $a$ with $a//1$ for each integer $a:a\equiv a//1$.

We now define a new operation on the rationals: reciprocal. If $x=a//b$ is a non-zero rationl then we define the reciprocal $x^{-1}$ of $x$ to be the rational number $x^{-1}:=b//a$.

Proposition 3.3 (Laws of algebra for rationals) Let $x,y,z$ be integers. Then we have \[ \begin{aligned} x+y&=y+x\\ (x+y)+z&=x+(y+z)\\ x+0=0+x&=x\\ x+(-x) = (-x)+x&=0\\ xy&=yx\\ (xy)z&=x(yz)\\ x1=1x&=x\\ x(y+z)&=xy+xz\\ (y+z)x=yx+zx \end{aligned} \] If $x$ is non-zero, we also have \[xx^{-1} = x^{-1}x=1.\]

The above set of ten identities are asserting that the rationals $\mathbb{Q}$ form a field.

We can now define the quotient of thwo rational numbers $x$ and $y$, provided that $y$ is non-zero, by the fomula \[ x/y:=x\times y^{-1}. \]

Definition 3.7 A rational number $x$ is said to be positive iff we have $x=a/b$ for some positive integers $a$ and $b$. It is said to be negatve iff we haave $x=-y$ for some positive rational $y$.

Lemma 3.3 (Trichotomy of rationals) Let $x$ be a rational number. Then exactly one of the following three statements is true: (a) $x$ is zero; (b) $x$ is a positive rational number; or (c) $x$ is a negative rational number.

Definition 3.8 (Ordering of the rationals) Let $x$ and $y$ be rational numbers. We say that $x>y$ iff $x-y$ is a positive rational number, and write $x<y$ iff $x-y$ is a negative rational number. We write $x\geq y$ iff either $x>y$ or $x=y$, and similarly define $x\leq y$.

Proposition 3.4 (Basic properties of order on the rationals) Let $x,y,z$ be rtional numbers. Then the following properties hold.

(Order trichotomy) Exactly one of the three statements $x=y$, $x<y$, $x>y$ is true.
(Order is anti-symmetric) One has $x<y$ iff $y>x$.
(Order is transitive) If $x<y$, then $x<y$ and $y<z$, then $x<z$.
(Addition preserves order) If $x<y$, the $x+z<y+z$.
(Positive multiplication preserves order) If $x<y$ and $z$ is positive, then $xz<yz$.

3.3 Absolute value and exponentiation

Definition 3.9 (Absolute value) If $x$ is rational number, then absolute value $|x|$ of $x$ is defined as follows. If $x$ is positive,then $|x|:=x$. If $x$ is negative,then $|x|:=-x$. If $x$ is zero, then $|x|:=0$.

Definition 3.10 (Distance) Let $x$ nd $y$ be rational numbers. Then quantity $|x-y|$ is called the distance between $x$ and $y$ and is sometimes denoted $d(x,y)$.

Proposition 3.5 (Basic properties of absolute value and distance) Let $x,y,z$ be rational numbers.

(Non-degeneracy of absolute value) We have $|x|\geq 0$. Also, $|x| = 0$ iff $x$ is 0.
(Triangle inequality for absolute value) We have $|x+y|\leq |x|+|y|$.
We have the inequalities $-y\leq x\leq y$ iff $y\geq |x|$.
(Multiplicativity of absolute value) We have $|xy|=|x||y|$.
(Non-degeneracy of distance) We have $d(x,y)\geq 0$. Also, $d(x,y)=0$ iff $x=y$.
(Symmetry of distance) $d(x,y)=d(y,x)$.
(Triangle inequality for distance) $d(x,z)\leq d(x,y)+d(y,z)$.

Definition 3.11 ($\varepsilon$-closeness) Let $\varepsilon$ be a rational number, and let $x,y$ be rational number. We say that $y$ is $\varepsilon$-close to $x$ iff we have $d(x,y)<\varepsilon$.

Proposition 3.6 Let $\varepsilon,\delta>0$. If $x$ and $y$ are $\varepsilon$-close, and $z$ and $w$ are $\delta$-close, then $xz$ are $yw$ are $\varepsilon|z|+\delta|x+\varepsilon\delta$-close.

Definition 3.12 (Exponentiation to a natural number) Let $x$ be a rational number. To raise $x$ to the power $0$, we define $x^0:=1$; in particular we define $0^0:=1$. Now suppose inductively that $x^n$ has been defined for some natural number $n$, then we define $x^{n+1} : = x^n\times x$.

Definition 3.13 (Exponentiation to a negative number) Let $x$ be a non-zero rational number. Then for any negative integer $-n$, we define $x^{-n} := 1/x^n$.

Proposition 3.7 (properties of exponentiation) Let $x,y$ be non-zero rational numbers, and $n,m$ be integers.

We have $x^nx^m = x^{n+m}$, $(x^n)^m = x^{nm}$, and $(xy)^n=x^ny^n$.
If $x\geq y>0$, then $x^n\geq y^n>0$ if $n$ is positive, and $0<x^n<y^n$ if $n$ is negative.
If $x,y > 0$, $n\ne 0$, and $x^n=x^y$, then $x=y$.
We have $|x^n| = |x|^n$.

3.4 Gaps in the rational numbers

Proposition 3.8 (Interspersing of integers by rationals) Let $x$ be rational number. Then there exists an integer $n$ such that $n\leq x<n+1$. In fact, this integer is unique. In poticular, there exists a natural number $N$ such that $N>x$.

The integer $n$ is somtimes referred to the integer part of $x$ and is sometimes denoted $n=[x]$.

Proposition 3.9 (Interspersing of rationals by rationals) If $x$ and $y$ are two rationals such that $x<y$, then there exists a third rational $z$ such that $x<z<y$.

There does not exist any rational number $x$ for which $x^2=2$.

To prove the proposition above, we need to a conclusion from the exercise 3.2.

For every rational numbers $\varepsilon>0$, there exists a non-negative rational number $x$ such $x^2<2<(x+\varepsilon)^2$.

Exercise 3.2 Prove the principle of infinite descent: that it is not possible to have a sequence of natural numbers which is in infinite descent (a sequence $a_0,a_1,\cdots$ of numbers is said to be infinite descent if we have $a_n>a_{n+1}$ for all natural numbers $n$).